Question: What is the period of $y=7\sin\left(-\dfrac{3\pi}{4} x-\dfrac{\pi}{4}\right)+6$ ? Give an exact value. units
Period in sinusoids of the form $y=a\sin(bx+c)+d$ Graphically, the period of a sinusoidal function is the horizontal distance between the ends of a single cycle of its graph. The period of a sinusoid of the form $y={a}\sin( {b}x + c) + {d}$ is equal to $\dfrac{2\pi}{| b|}$. [How can we justify this given our graphical understanding of period?] Finding the period The period of $y = 7\sin\left({-\dfrac{3\pi}{4}}x-\dfrac{\pi}{4}\right)+6$ is: $\begin{aligned} \text{period}&=\dfrac{2\pi}{|{b}|}\\\\ &=\dfrac{2\pi}{\left|{-\dfrac{3\pi}{4}} \right|} \\\\\\\\\\ &=\dfrac{2\pi}{{\dfrac{3\pi}{4}}}\\\\\\\\\\ &= 2\pi\cdot\dfrac{4}{3\pi}\\ \\ &=\dfrac83 \end{aligned}$ The answer The period of $y = 7\sin\left({-\dfrac{3\pi}{4}}x-\dfrac{\pi}{4}\right)+6$ is $\dfrac83$ units.